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LieAlgebras[MultiplicationTable] - display the multiplication table of a Lie algebra or a general non-commutative algebra

Calling Sequences

     MultiplicationTable(LieAlgebraName, keyword)

Parameters

     LieAlgebraName  - (optional) name or string, the name assigned to a Lie algebra

     keyword         - keyword string, one of "LieBracket", "ExteriorDerivative", "LieDerivative", "AlgebraTable"

 

Description

Examples

Description

• 

MultiplicationTable(LieAlgebraName, keyword) displays the form of structure equations for the Lie algebra or algebra dictated by the keyword.

• 

If the keyword is "LieBracket", then the Lie brackets ei, ej of the basis elements e1, e2, ..., en are displayed in a two-dimensional array.

• 

If the keyword is "AlgebraTable", then the non-commutative products eiej of the basis elements e1, e2, ..., en are displayed in a two-dimensional array.

• 

If the keyword is "ExteriorDerivative", then the exterior derivatives dθiof the dual basis elements θ1, θ2, ... , θn are printed.

• 

If the keyword is "LieDerivative", then the Lie derivatives ℒeiθj of the dual 1-forms θ1, θ2, ... , θn with respect to the basis vectors e1, e2, ..., en are displayed in a two-dimensional array.

• 

If LieAlgebraName is omitted, then the appropriate multiplication table of the current algebra is displayed.

• 

The command MultiplicationTable is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form MultiplicationTable(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MultiplicationTable(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we initialize a 5 dimensional Lie algebra.

L1_DGLieAlgebra,Alg1,5,2,3,1,1,2,5,3,1,4,5,4,1:

DGsetupL1:

 

Display the Lie bracket multiplication table.

Alg1 > 

MultiplicationTableLieBracket

e2,e3=e1,e2,e5=e3,e4,e5=e4

(2.1)

 

Display the exterior derivatives of the dual 1-forms.

Alg1 > 

MultiplicationTableExteriorDerivative

dθ1=θ2θ3

dθ2=0θ1θ2

dθ3=θ2θ5

dθ4=θ4θ5

dθ5=0θ1θ2

(2.2)

 

Display the Lie derivatives of the dual 1-forms.

Alg1 > 

MultiplicationTableLieDerivative

| θ1θ2θ3θ4θ5------------------------e1| 00000e2| θ30θ500e3| θ20000e4| 000θ50e5| 00θ2θ40

(2.3)

 

Example 2.

We initialize a 4 dimensional Lie algebra. Instead of using the standard default labels for the basis vectors we use X, Y, U,V and for the dual 1-forms we use α, β, σ, τ.

Alg1 > 

L2_DGLieAlgebra,Alg1,4,2,3,1,1,2,4,3,1,4,2,4,1:

Alg1 > 

DGsetupL2,X,Y,U,V,α,β,σ,τ:

 

Display the Lie bracket multiplication table.

Alg1 > 

MultiplicationTableLieBracket

e2,e3=e1,e2,e4=e3

(2.4)

 

Display the exterior derivatives of the dual 1-forms.

Alg1 > 

MultiplicationTableExteriorDerivative

dα=βσ

dβ=0αβ

dσ=βτ

dτ=βτ

(2.5)

 

Display the Lie derivatives of the dual 1-forms.

Alg1 > 

MultiplicationTableLieDerivative

| αβστ--------------------X| 0000Y| σ0ττU| β000V| 00ββ

(2.6)

 

Example 3. 

We initialize the quaternions ℍ and display the multiplication table.

   

Alg1 > 

L3AlgebraLibraryDataQuaternions,H

L3:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e2.e1=e2,e22=e1,e2.e3=e4,e2.e4=e3,e3.e1=e3,e3.e2=e4,e32=e1,e3.e4=e2,e4.e1=e4,e4.e2=e3,e4.e3=e2,e42=e1

(2.7)
Alg1 > 

DGsetupL3,e,i,j,k,θ

algebra name: H

(2.8)
Alg1 > 

MultiplicationTableAlgebraTable

| eijk---- ---- ---- ---- ---- e| eijki| iekjj| jkeik| kjie

(2.9)

See Also

DifferentialGeometry

LieAlgebras

ExteriorDerivative

LieBracket

LieDerivative